Math 251 - Alexiades
Review for Exam 2 on Chap.4   printable (.pdf)
Concepts
  • vector space, subspace
  • linear independence, span, basis, dimension
  • row space, column space, nullspace, rank, nullity of m×n matrix
  • linear transformation T: V → W
  • one-to-one, onto transformations

    Precise definition of
  • vector space (10 axioms), subspace: a subset of a v.s. which is itself a v.s. with the same operations.
  • linearly independent set {v1,v2,...,vk} in V :  c1v1+c2v2+...+ckvk=0 implies c1,c2,...,ck are all 0.
  • basis of V :  a subset of V which is linearly independent and spans V.
  • dimension of V :  the number of vectors in any basis of V.
  • linear transformation T: V → W :   T(c1x1+c2x2) = c1T(x1)+c2T(x2) ∀ scalars ci, ∀ xi∈V.
  • one-to-one mapping T: V → W :   T(u)=T(v) implies u=v  ∀u,v ∈ V.
         For a linear transformation: being one-to-one amounts to: T(x)=0 implies x=0,
                                        i.e. nullity(T)=0, i.e. rank[T] = n
  • onto mapping T: V → W :   for each y ∈ W, y = T(x) for some x ∈ V,
                            i.e. T(x) = y has a solution x ∈ V for each y ∈ W
         For a linear transformation: being onto amounts to: [T]x=y solvable ∀ y ∈ ℝm,
                                  i.e. cols of [T] span ℝm, i.e. rank[T] = m

    Methods: Know How To
  • decide if a set/subset (with given operations) constitutes a vector space/subspace
  • decide if a set {v1,v2,...,vk} is linearly independent
  • express a vector as a linear combination of {v1,v2,...,vk}
  • decide if a set {v1,v2,...,vk} spans a space/subspace
  • find a basis (and dimension) for a space/subspace
  • find a basis for row space, column space, nullspace of m×n matrix
  • decide if a transformation T: V → W is linear or not
  • find standard matrix for a linear transformation T: ℝn → ℝm
  • decide if a linear transformation T: V → W is one-to-one, if it is onto

    Basic / crucial Theorems to know (...among others...)
  • Dimension Thm:   rank(A) + nullity(A) = n

  • What rank tells us, for an m×n matrix A:
              rank(A) = r   ⇐⇒ the column space of A is an r-dimensional subspace of ℝm
                  ⇐⇒ r of the n columns are linearly independent in ℝm
                  ⇐⇒ the row space of A is an r-dimensional subspace of ℝn
                  ⇐⇒ r of the m rows are linearly independent in ℝn
                  ⇐⇒ nullity(A) = n−r     ⇐⇒     Ax=0 has n−r linearly independent solutions
              rank(A) = n   ⇐⇒ nullity(A) = 0     ⇐⇒     Ax=0 has only the trivial solution
                  ⇐⇒ the columns of A are linearly independent in ℝm
                  ⇐⇒ Ax=b has at most one solution ∀ b ∈ ℝm
              rank(A) = m   ⇐⇒ the columns of A span ℝm   ⇐⇒ Ax=b is consistent ∀ b ∈ ℝm

  • Fundamental Thm for square matrices:   For a square n×n matrix A, the following are equivalent:
       1.  A is invertible
       2.  A ∼ In   (i.e. A is row equivalent to the Identity)
       3.  A is expressible as a product of elementary matrices
       4.  rank(A) = n
       5.  det(A) ≠ 0
       6.  the homogeneous system Ax = 0 has only the trivial solution
       7.  nullity(A) = 0
       8.  the nonhomogeneous system Ax = b is consistent ∀ b ∈ ℝn
       9.  the nonhomogeneous system Ax = b has unique solution ∀ b ∈ ℝn
        10.  λ = 0 is not an eigenvalue of A
        11.  the columns [and rows] of A are linearly independent vectors in ℝn
        12.  the columns [and rows] of A span ℝn
        13.  the columns [and rows] of A form a basis for ℝn
        14.  the linear operator TA: ℝn → ℝn, defined by TA(x)=Ax, is one-to-one
        15.  the linear operator TA: ℝn → ℝn, defined by TA(x)=Ax, is onto